The category of affine algebraic regular monoids

The main aim of this study is to characterize affine weak $k$-algebra $H$ whose affine $k$-variety $S=M_k(H,k)$ admits a regular monoid structure. As preparation, we determine some results of weak Hopf algebras morphisms, and prove that the anti-function from the category $\mathcal{C}$ of weak Hopf algebras whose weak antipodes are anti-algebra morphisms is adjoint. Then, we prove the main result of this study: the anti-equivalence between the category of affine algebraic $k$-regular monoids and the category of finitely generated commutative reduced weak $k$-Hopf algebras.